Arnold Mathematical Journal (2020) 6:1–20https://doi.org/10.1007/s40598-020-00132-0

PROBLEM CONTRIBUT ION

On Convex Bodies that are Characterizable by VolumeFunction

“Old and Recent Problems for a New Generation”: A Survey

Ákos G. Horváth1

Received: 29 July 2019 / Revised: 3 January 2020 / Accepted: 8 January 2020 / Published online: 23 January 2020© The Author(s) 2020

AbstractThe “old-new” concept of a convex-hull function was investigated by several authorsin the last seventy years. A recent research on it led to some other volume functions asthe covariogram function, the widthness function or the so-called brightness functions,respectively. A very interesting fact that there are many long-standing open problemsconnected with these functions whose serious investigation was closed before the “ageof computers”. In this survey, we concentrate only on the three-dimensional case; wewill mention the most important concepts, statements, and problems.

Keywords Centred convex body · Central symmetric convex body · Convex-hullfunction · Covariogram function · Difference body · Blaschke body

Mathematics Subject Classification 52A40 · 52A38 · 26B15 · 52B11

1 Introduction

The “old-new” concept of a convex-hull function was investigated by several authorsin the last seventy years. A recent research on it led to some other volume functions asthe covariogram function, the widthness function or the so-called brightness functions,respectively. A very interesting fact is that there aremany long-standing open problemsconnected with these functions whose serious investigation was closed before the “ageof computers”. The structure of the conjectured optimal bodies reflects quite a theoret-ical attitude: seemingly, there was no computer research to support them. In this paper,we collect some of them (using the necessary theoretical knowledge) to inspire thecomputer experts for such research which can reorder the map of these problems. We

B Ákos G. Horváthghorvath@math.bme.hu

1 Department of Geometry, Mathematical Institute, Budapest University of Technologyand Economics, Budapest 1521, Hungary

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2 A. G. Horváth

concentrate on the three-dimensional case; we mention the most important concepts,statements, and problems.

We use the following notation. K is a convex body; the class of convex bodies isdenotedbyK. K+L denotes theMinkowski sumof the convexbodies K and L and−Kis the reflected image of K with respect to the origin. Rn and Sn−1 are the analyticmodels of the n-dimensional Euclidean space and the (n − 1)-dimensional sphere,respectively. A convex body with continuously differentiable boundary is called a C1-body; analogously, a convex body with boundary of positive Gauss curvature is calleda C2+-body. We use the following special notation:

• DK : the difference body of K• �K : the central symmetral of K• �K : the projection body of K• �K : the Blaschke body of K• gK (u): the covariogram function of K• hK (u): the support function of K• wK (u): the width function of K• bK (u): the brightness function of K• GK (u): the convex-hull function of K• G(u): the Gauss map.

From the general theory of convex sets, we use the concept of Minkowski normgenerated by a convex body centrally symmetric with respect to the origin; the conceptof polar body and the special 2-dimensional norm, the so-called Radon norm with theboundary of its unit disk being a Radon curve. The following statements are importantin our arguments:

• Brunn–Minkowski’s inequality: If K and L are convex bodies then

voln (K + L)1n ≤ voln (K )

1n + voln (L)

1n

• Alexandrov’s projection theorem: Let 1 ≤ i ≤ k ≤ n − 1, and let K , L becentrally symmetric compact convex sets of dimension at least i + 1 in R

n . IfVi (K |S) = Vi (L|S) for all k-dimensional subspaces S ofRn , then K is a translateof L .

• Cauchy’s projection formula:

V (K |u⊥) = 1

2

∫

Sn−1

〈u, u〉dS(K , v).

• Minkowski’s existence theorem: For the finite Borel measure μ in Sn−1 to beSn−1(K , ·) for some convex body K ∈ K, it is necessary and sufficient that μ isnot concentrated on any great subsphere of Sn−1, and

∫

Sn−1

udμ(u) = 0.

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On Convex Bodies that are Characterizable. . . 3

• Minkowski’s first inequality: Let V n(K , n − 1; L) be the mixed volumeV n(K , . . . , K , L) where the number of the copies of K is n − 1; then

V n(K , n − 1; L) ≥ voln−1n (K ) voln(L),

with equality if and only if K and L lie in parallel hyperplanes or are homothetic.

2 The Covariogram Function

Let K be a convex body in Rn . The covariogram gK of K is the function

gK (t) := voln(K ∩ (K + t)), (1)

where t ∈ Rn and voln denotes n-dimensional volume. This functional, which was

introducedbyMatheron inhis bookMatheron (1986) on randomsets, is also sometimescalled the set covariance and it coincides with the autocorrelation of the characteristicfunction 1K of K :

gK = 1K �1−K .

The covariogram gK is clearly unchanged by a translation or a reflection of K .(The term reflection will always mean reflection in a point.) Matheron (1986) askedthe following question and conjectured a positive answer for the case n = 2.

Problem 1 (Covariogram problem) Does the covariogram determine a convex bodyamong all convex bodies up to translations and reflections?

The first contribution to Matheron’s question was made by Nagel (1993), whoconfirmedMatheron’s conjecture for all convex polygons. Other partial results towardsthe complete confirmation of this conjecture in the plane have been proved by Schmitt(1993), Bianchi et al. (2002), Bianchi (2005), Averkov and Bianchi (2007). In general,the answer to the covariogram problem is negative, as the author of Bianchi (2005)proved by finding counterexamples in R

n for any n ≥ 4. Indeed, the covariogramof the Cartesian product of convex sets K ⊂ R

k and L ⊂ Rm is the product of the

covariograms of K and L . Thus K × L and K × (−L) have equal covariograms.However, if neither K nor L is centrally symmetric, then K × L is neither a translationnor a reflection of K × (−L). To satisfy these requirements, the dimensions of bothsets must be at least two and thus the dimension of the counterexamples must be atleast four.We note that these counterexamples can be polytopes but notC1 bodies. Forn-dimensional convex polytopes P , Goodey et al. (1997) proved that if P is simplicialand P and−P are in general relative position (the polytope is a generic polytope), thecovariogram determines P . Finally, Bianchi (2009) proved that for three-dimensionalpolytopes the conjecture is also true. We note that the general case of dimensionthree is open: it is still not known whether every three-dimensional convex body isdetermined by its covariogram. We collected the most important statements in thefollowing theorem:

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4 A. G. Horváth

regular hexagon

Cuboctahedron

Fig. 1 The difference bodies of simplices

Theorem 1 Let K be a convex body of dimension n in the Euclidean n-space.

• The covariogram function determines a convex body K among all convex bodiesup to translations and reflections in the following cases:

• K is centrally symmetric,• dim K = 2,• K is a polytope, and dim K = 3,• K is a generic polytope of dimension n.

• If n ≥ 4, there are convex polytopeswhich are not determined by their covariogramfunction.

We first deal with such bodies which are related to this nice problem.

2.1 The Difference Body and the Central Symmetral of K

First observe that the covariogram function gK determines the volume of K , sincegK (0) = voln(K ). What the covariogram obviously does determine is its support,which is the convex body

DK := {x ∈ Rn : K ∩ (K + x) �= ∅} = K − K . (2)

This is the difference body of K. Sometimes is more convenient to define the so-called central symmetral �K of K which is the homothetic by factor 1/2 copy ofthe difference body. The difference body can be constructed as follows. Supposethat the origin o is in the interior of K , and take the union of all translates of −Kwhich are placed so that the corresponding translate of o lies on the boundary of K .Finally, if we dilate by a factor 1/2, we get �K . The central symmetral of a regulartriangle is a regular hexagon, and the central symmetral of a regular tetrahedron isthe cuboctahedron (see Fig. 1). Since for a centrally symmetric convex body K with

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On Convex Bodies that are Characterizable. . . 5

center c, the equality −K = K − 2c holds,

t K + (1 − t)(−K ) = K − 2c(1 − t),

and in particular �K is a translate of K . Hence a centrally symmetric convex body isdetermined (among all convex bodies) by its covariogram, since 2K is a translated copyof DK . In general, �K does not determine K meaning that there are non-congruentconvex bodies with the same central symmetral.

2.2 The Support Function and theWidth Function

The support function hK of K is defined by

hK (x) = max{〈x, y〉 : y ∈ K } (3)

for x ∈ Rn . As a function, the support function is positively homogeneous and subad-

ditive, that is, a sublinear function. It can be observed that a convex body is determinedby its support function [see (0.6) in Gardner (1995)].

Let K be a compact, convex set in Rn . Then K has two supporting hyperplanes,

which are orthogonal to a unit vector u. The distance between these hyperplanes isthe width wK (u) of K in the direction of u. The width of K is the maximal value ofthe width function; the thickness of K is the minimal one, respectively. The formaldefinition of the width function is

wK (u) := hK (u) + hK (−u) (4)

for u ∈ Sn−1. Since

wt K+(1−t)K = twK + (1 − t)w−K = wK ,

there is a whole continuum of non-congruent compact convex sets, which have thesame width function. One of them is the central symmetral of K , which shows thatw�K = wK for all convex compact bodies. (Generally, �K is the unique centeredcompact convex set with this property.)

The support function of the central symmetral is equal to h�K = 1/2(hK +h−K ) =1/2wK ; thus the support function of the difference body is the width function of K.In particular, the covariogram of a convex body determines its width function, but thisproperty also allows a great uncertainness in the determination of K . We can definethe class of convex bodies with constant width by the equation

wK (u) = hK (u) + hK (−u) = constant. (5)

Clearly, the n-dimensional ball is of constant width. On the plane, nonspherical exam-ples are the Reuleaux polygons, and there is an analogous theorem in any dimension.Precisely, it can be proved that nonspherical convex bodies of constant width exist

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6 A. G. Horváth

in Rn for all n ≥ 2 [see Theorem 3.2.5 in Gardner (1995)]. There are nice inves-

tigations of this class of bodies. Evidently, a convex body K is of constant widthif and only if �K is a ball. If K is a centrally symmetric convex body in R

n , thenhK (u) = h−K (u) = hK (−u) implying that wK (u) = hK (u) + h−K (u) = 2hK (u).So if the width function of a centrally symmetric convex body is constant, then itssupport function is also constant; hence the body must be a ball. Using the Brunn–Minkowski inequality [see Theorem 7.1.1 in Schneider (2014)], we get that in theclass of convex bodies of constant width d, the ball has the largest volume. In fact, thecentral symmetral of K is �K = 1/2(K + (−K )) with the same width as K , but it iscentrally symmetric; hence it is a ball with the same width function as K . On the otherhand, the Brunn–Minkowski inequality gives that for the central symmetral (which isthe ball of the same class of constant width)

voln(�K )1n = voln

(1

2(K + (−K ))

) 1n ≥ voln(K )

1n ,

as we stated. Now the question “Whose body has the minimal volume in a classof convex bodies of constant width d?” is natural. In the class of all planar convexsets of constant width d, the Reuleaux triangle has the least area. The first proofsof this theorem are contained in the papers by Lebesgue (1914) and by Blaschke(1915). Similar question was investigated by Pal (1921) who showed that the regulartriangle has the least area among all convex sets of given thickness d. Recently, Campi,Colesanti and Gronchi investigated the minimum volume problem in Campi et al.(1996). They considered the 3-dimensional case and raised the following problems:

Problem 2 (Campi–Colesanti–Gronchi) Find a convex body of the minimum volumein each of the following classes:

(A) The class of convex bodies of constant width d;(B) The class of convex bodies of thickness d.

Notice that the existence of solutions for each of these problems is guaranteed bystandard compactness arguments. Problems A and B are still unsolved. On the otherhand, several authors turned their attention to the classes of convex bodies involvedin those problems [for a detailed history read the paper Campi et al. (1996)]. Weconsider only those bodies which are good candidates to solve these problems. Firstwe mention here the Reuleaux tetrahedron which can be obtained from a regulartetrahedron with the edge of length d, as the intersection of four balls centred at thevertices of the tetrahedron with radius d. Unfortunately, Reuleaux tetrahedron is not abody of constant width: the midpoints of its two opposite curved edges have distancegreater than d. Meissner and Schilling (1912) showed how to modify the Reuleauxtetrahedron to form a body of constant width by replacing three of its edge arcs bycurved patches formed as the surfaces of rotation of a circular arc. Incidentally, asMeissner mentioned on p. 49 of Meissner (1911), the ball is the only body of constantwidth that is bounded only by spherical pieces. According to this, if three edge arcsare replaced (three that have a common vertex or three that form a triangle), twononcongruent shapes arising are called Meissner tetrahedra (see Figs. 2, 3, 4).

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On Convex Bodies that are Characterizable. . . 7

A

A

B

B

C

CD

D

Fig. 2 The Meissner tetrahedron

Fig. 3 The body of revolution with the minimal volume

Bonnesen and Fenchel conjectured in Bonnesen and Fenchel (1934):

Conjecture 1 (Bonnesen and Fenchel 1934) Meissner tetrahedra are the minimum-volume three-dimensional shapes of constant width.

This conjecture is still open. (I propose to read the nice paper of Kawohl andWeber(2011) on this conjecture.) In connection with this problem, Campi, Colesanti andGronchi showed.

Theorem 2 (Campi et al. 1996) The minimum volume surface of revolution with con-stant width is the surface of revolution of a Reuleaux triangle through one of itssymmetry axes.

A candidate to solve Problem B was proposed by Heil (1978); in this case theconstruction is based upon a tetrahedron, too. Namely, the Heil body is the convexhull of six circular arcs of radius d, centered at the mid-points of the edges of a regulartetrahedron with edges of length d

√2, and the four vertices of a rescaled tetrahedron

with edges of length d(2√6 − √

2)/3.

Conjecture 2 (Heil 1978) Heil body is the minimum-volume three-dimensional bodyin the class of convex bodies with given thickness.

Campi, Colesanti and Gronchi solved this problem for surfaces of revolution usingthe concept of a shaken body. Fix a plane H orthogonal to u ∈ Sn−1 and a closed

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8 A. G. Horváth

Fig. 4 The Heil body

half-space H+ bounded by H . The shaken body SK (u) of K with respect to u is the setcontained in H+ such that for every line l parallel to u, SK (u) ∩ l is either a segmenthaving an endpoint on H and the same length as K ∩ l, or the empty set, whether lintersects K or not [see Bonnesen and Fenchel (1934)]. Clearly, the shaken body is aconvex body of the same volume as K . On the other hand, Campi at al. proved that thewidth function of the shaken body in the direction of the axis of a three-dimensionalconvex body of revolution is greater than the width function of K . From this the authorshowed

Theorem 3 (Campi et al. 1996) Among all three-dimensional convex bodies of rev-olution of a given thickness d, the unique body of the minimum volume is the conegenerated by the revolution of a regular triangle with side 2d/

√3 around one of its

axes of symmetry.

2.3 The Brightness Function, the Projection Body, and the Blaschke Body

The brightness function of K is also determined by the covariogram function. For adisk, it can be obtained by the rotation of the width function around the origin by π/2.The brightness of K in the direction of the unit vector u is the (n − 1)-dimensionalvolume of the orthogonal projection of K to a hyperplane with the unit normal vectoru. The brigthness function is the function

bK (u) : u �→ voln−1(K |u⊥), (6)

where (K |u⊥) is the orthogonal projection of K onto the hyperplane with the normalvector u. Our notes on the brigthness function of a disk is clear from Fig. 5. Thecovariogram function determines the brightness function of the body. In fact, foru ∈ Sn−1,

limr �→0+0

d

drgK (ru) = − voln−1(K |u⊥) = −bK (u). (7)

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On Convex Bodies that are Characterizable. . . 9

u

(u)

(u)

K

K

w

b K

Fig. 5 The width function and the brightness function of a disk

nn

nn

nn

n

n

1

1

2

2

3

3

4

4

Fig. 6 The projection body of the regular tetrahedron

This follows from the facts that

r voln−1((K ∩ (K + ru))|u⊥) ≤ voln(K \ (K + ru)) ≤ r voln−1(K |u⊥),

and limr �→0+0 K ∩(K +ru) = K . The answer for the question whether the brightnessfunction determines or does not determine the body is known. Let us first introducethe projection body �K of K as the uniquely defined body whose support functionat the point u is equal to the value of the brightness function of K at u. If we restrictour investigations to centered convex bodies ofRn , then fromAlexandrov’s projectiontheorem [see Theorem 3.3.6 in Gardner (1995)] we get that if K1 and K2 have the samebrightness functions (or, equivalently, the projection bodies �K1 and �K2 agree),then K2 is a translate of K1. In general, as we will see later, there is a continuumof (generally non-congruent) sets with the same brightness function as K with thisproperty.

On Fig. 6, we can see the projection body of the regular tetrahedron. To determinethe projection body of a polyhedron, we can use the Cauchy projection formula [seeA.45 in Gardner (1995)], which relates the volumes of projections to the surface areameasure as follows:

h�K (u) := bK (u) = voln−1(K1|u⊥) = 1

2

∫

Sn−1

|〈u, v〉|dSn−1(K , v). (8)

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10 A. G. Horváth

The surface area measure of the regular tetrahedron is the measure concentrated tofour vertices of a regular tetrahedron inscribed in the unit sphere. These points haveequal measures (see ni on the figure). The integral then reduces to the sum of thefour terms |〈u, ni 〉|, i = 1, . . . , 4. Each term is the support function of a line segment[0, voln−1(Fi )ni ], where Fi is the facet orthogonal to ni . By the property of the supportfunction, the projection body is the Minkowski sum of the four segments. Hence �Kis a rhombic dodecahedron (see on the right of Fig. 6).

Let K be a convex body of dimension n. The Gauss map G sends a unit normalvector of the boundary of K to the corresponding point of the sphere Sn−1. We candefine a measure on Sn−1 by means of the Gauss map. For a set H ⊂ Sn−1, weconsider the set G−1(H) of those points of K , in which the unit normal vector ismapped by the Gauss map to a point of H . Let the measure of H be the surface areaof the set G−1(H). We denote by Sn−1(K , ·) this surface area measure. Minkowki’sexistence theorem [see A.3.2. in Gardner (1995)] says that for a finite Borel measureon the unit sphere not concentrated on a great subsphere of Sn−1 there is a convexbody K , for which the surface area measure Sn−1(K , ·) is the given one. Hence if Kis a convex body in R

n , and 0 ≤ t ≤ 1, then there is a unique convex body whosesurface area measure is (1 − t)Sn−1(K , ·) + t Sn−1(−K , ·). The relation between thebrightness function and the surface area measure of K is the following: The conditionthat for all u ∈ Sn−1, voln−1(K1|u⊥) = voln−1(K2|u⊥) is equivalent to that for allu ∈ Sn−1, Sn−1(K1, ·)+ Sn−1(−K1, ·) = Sn−1(K2, ·)+ Sn−1(−K2, ·) [see Theorem3.3.2 in Gardner (1995)]. This leads to another important observation: since for all0 ≤ t ≤ 1

((1 − t)Sn−1(K , ·) + t Sn−1(−K , ·)) + ((1 − t)Sn−1(−K , ·) + t Sn−1(K , ·))= Sn−1(K , ·) + Sn−1(−K , ·),

the unique convex body, whose surface area measure is (1 − t)Sn−1(K , ·) +t Sn−1(−K , ·), has the same brightness function as K . This also implies that theirprojection bodies are the same centred convex body.

The Blaschke body �K corresponds to t = 1/2. The term projection class of K isused for the class of all convex bodies K ′ for which �K = �K ′. �K is the uniquebody in a projection class with the largest volume [see Theorems 4.1.3 and 3.3.9 inGardner (1995)]. From the proof [using Minkowski’s first inequality, B.11 in Gardner(1995)], it follows that the volume of K is equal to the volume of �K if and onlyif K is centrally symmetric. This means that the only centrally symmetric elementof a projection class is the Blaschke body. In Fig. 7, we drew the Blaschke body ofthe regular tetrahedron. The surface area measure of −K is the reflected image of thesurface area measure of K . Hence the surface area measure of the Blaschke body isconcentrated to eight points with equal masses situated at the outward unit vectors ofthe facets of a regular octahedron; hence the body�K is the regular octahedron, whichis the intersection of the regular tetrahedron K and its reflected image −K (here theorigin is the common centroid of the two tetrahedra).

In dimension 3, a convex body has constant brightness if its shadow on any planealways has the same area. Among centrally symmetric convex bodies, the ball is the

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On Convex Bodies that are Characterizable. . . 11

n n

nn

nn

n

n

1 1

2

2

33

4

4

Fig. 7 The Blaschke body of the regular tetrahedron

Fig. 8 A nonspherical body ofconstant brightness

u

-u

u1

4

44

44

only one with constant brightness [Theorem 3.3.11 in Gardner (1995)]. Blaschke con-structed the first example of a nonspherical convex body in R3 of constant brightness(see Blaschke 1949). He observed that such a body can be found in the class of solidsof revolution with the z-axis as the axis of revolution. Let us briefly describe this niceconstruction. It is based on the fact that a convex body of class C2+ has a constantbrightness if and only if the sum of the products of the principal radii at antipodalpoints is constant [see Theorem 3.3.14 in Gardner (1995)]. Blaschke described the2-dimensional meridian section of the body K , which lies in the {x, z}-plane. Let xube the point of this meridian, where the outer normal vector is u. The directions corre-sponding to the principal radii of Gauss curvature are {u1, u2}. We take u1 in the planeof {u, z}, orthogonal to u and u2 orthogonal to the plane {u, u1}. The second principalradius R2 is the distance from xu to the z-axis measured along the line through xuparallel to u. We need

R1(u)R2(u) + R1(−u)R2(−u) = const = c

for all u. We can see the examined meridian in Fig. 8. Note that the base point of oneof two opposite normal vectors is in the vertex of this meridian. Hence the smooth

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12 A. G. Horváth

parts have to be of constant Gauss curvature with the same value (in a point of non-smoothness, the Gauss curvature is infinity given that one of the principal radii is equalto zero). For a surface of revolution of the form

F(x, ϕ) = (x cosϕ, x sin ϕ, z(x)),

the Gauss curvature K is known to be

1

R1(u)R2(u)= K = z z

x(1 + z2)2.

By the substitution t = z2, t = 2z z, we get from this in the case of K �= 0 thedifferential equation

t

(1 + t)2= 2Kx .

The general solution for t is

z2 = t = −1 + c + Kx2

c + Kx2,

where c is an arbitrary constant. In our case of K = 1, we also know the initialcondition: z(0) = −1. Hence c = −1/2, and

z(x) = ±x∫

0

√1 + 2x2

1 − 2x2dx + C

with another constant C . Substituting cos v = √2x in this non-elementary integral

we get

z

(cos v√

2

)= ∓ 1√

2

cos−1√2x∫

π/2

√1 + cos2 vdv + C = ± 1√

2

π/2∫

cos−1√2x

√2 − sin2 vdv + C .

The initial condition now is z(√2/2) = 0; hence

C = ∓ 1√2

π/2∫

0

√2 − sin2 vdv.

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On Convex Bodies that are Characterizable. . . 13

Since z(v) ≥ 0, and 1√2

π/2∫0

√2 − sin2 vdv ≥ ± 1√

2

π/2∫cos−1

√2x

√2 − sin2 vdv, we have

to choose the positive sign for C . Hence the only possibility is

z

(√2

2cos v

)=

√2

2

⎛⎜⎝

π/2∫

0

√2 − sin2 vdv −

π/2∫

cos−1√2x

√2 − sin2 vdv

⎞⎟⎠

=√2

2

cos−1√2x∫

0

√2 − sin2 vdv,

which is the meridian proposed by Blaschke in Blaschke (1949).A nice problem arose in the common investigation of width and brightness. In the

plane, the two concepts are essentially the same and they give the same information onthe body. In 3-space, this is not true; so we have the following question: Is every convexbody in R3 of constant width and constant brightness a ball? The positive answer forC2 bodies was given by Nakajima (1926). He showed that any convex body inR3 withconstant width, constant brightness, and the boundary of class C2 is a ball. In 2005,Howard proved in Howard (2006) that the regularity assumption on the boundary isunnecessary, so balls are the only convex bodies of constant width and brightness.

Turning back to paper of Campi at al., we find the following problems:

Problem 3 (Campi–Colesanti–Gronchi) Find a convex body of the minimum volumein each of the following classes:

(C) The class of convex bodies of constant brightness b;(D) The class of convex bodies of the minimal brightness b.

To get a partial solution of problem C), the authors proved the following

Theorem 4 (Campi et al. 1996) Let K be a body of the minimum volume in the classof all convex bodies of constant brightness b. Then the surface area measure S2(K , ·)of K has the following form:

S2(K , H) =∫

H

sK (z)dz, H is a Borel set in S2, (9)

where sK (·) is a non-negative function in L1(S2) such that

sK (z) + sK (−z) = −2b

πalmost ewerywhere in S2, (10)

sK (z)sK (−z) = 0 almost ewerywhere in S2. (11)

This theorem leads to further conjectures:

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14 A. G. Horváth

Conjecture 3 (Campi–Colesanti–Gronchi on Problem C) Consider three pairwiseorthogonal great circles on S2; they determine eight open regions. Define sK as apiecewise constant function whose values are 0 and 2b/π on those regions alter-nately. Let K be the unique convex body such that sK is the distribution of its surfacearea measure. Clearly, K has constant brightness b, and the Gauss curvature is π/2bat each regular point of K .

The authors conjectured that this body is a solution of Problem C.

Conjecture 4 (Campi–Colesanti–Gronchi on Problem C when the body is a body ofrevolution) The body constructed by Blaschke has the minimum volume among allconvex bodies of revolution with constant brightness b.

To support this conjecture, they noticed that Blaschke’s body of constant brightnesscan be constructed in the following manner: Fix on S2 a great circle as an equatorand take the pair of parallel circles at a spherical distance π/4 from it. In such a wayS2 is divided into four open regions. Define sK to be a piecewise constant functionwhose values are 0 and 2b/π on those regions alternately; sK is the distribution of thearea measure of the convex body of revolution with constant brightness constructedby Blaschke (1949). In the nice paper Gronchi (1998), proved this conjecture:

Theorem 5 (Gronchi 1998) The n-dimensional Blaschke–Firey body is the unique (upto a rigid motion) element of the minimal volume in the class of all convex bodies ofrevolution with constant brightness b.

We note that problem D) is still open, although the regular simplex is a strongcandidate to solve it.

3 The Convex-Hull Function

The convex-hull function of a convex body K is the “dual” of the covariogram function.It also has been examined for a long time. The first paper related to it was written byFáry and Rédei (1950). They proved that if one of the bodies moves on a line withconstant velocity, then the volume of the convex hull is a convex function of time [seeSatz.4 in Fáry and Rédei (1950)]. The general definition is the following:

Definition 1 Let K be an n-dimensional convex compact body; associate with a trans-lation vector t ∈ R

n the value GK (t) = vol conv{K ∪ (K + t)}. We call this functionthe convex-hull function, associated with the body K.

In dimension 2, this statement says that an intersection of the graph of the convex-hull function with a plane parallel to the z-axis is a convex function. Later, severalpapers related to this observation suggested a lot of interesting problems [see the surveyHorváth (2018) and the references therein]. It is interesting that the “basic problem”on the convex-hull function, whether the convex-hull function determines or does notdetermine the body K was asked only recently by Á. Kurusa.

To answer this question, we have to collect some general information on the convex-hull function. Obviously GK (0) = vol(K ), and its support is the whole Rn . Denote

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On Convex Bodies that are Characterizable. . . 15

P

P

Q I

Fig. 9 Non-congruent nonagons with the same brightness function and with the same volume

by u a unit vector and (as in previous sections) by K |u⊥ the orthogonal projection ofK onto a hyperplane with the normal vector u.

Lemma 1 Let α be an arbitrary real number, and u ∈ Sn−1 be a unit vector in Rn.

Then we have

GK (αu) = voln(K ) + |α| voln−1(K |u⊥); (12)

consequently, the function u �→ GK (u) − voln(K ) is the brightness function of K .Conversely, the brightness function of K and the value voln K determine the convex-hull function of K .

Proof The equality (12) is an easy consequence of the Cavalieri principle [see alsoSect. A.5 in Gardner (1995)]. The second statement follows from the definition of thebrightness function. The third statement is an immediate consequence of the equality(12). ��Remark In the case when the brightness function of K determines the body K , it alsodetermines the convex-hull function of K . On the other hand, we discussed earlierthat there are several bodies with the same brightness function. The only question isthat if the volume and the brightness function are both known, then whether the bodyis determined or not. Since the covariogram function also has these two properties indimension n for n ≥ 4, the answer should be “no”.We know that on the plane there aretwo non-congruent convex polygons with the same width function and the same area.To get an example, take two congruent regular hexagons and add (in a suitablemanner)to each of them three congruent suitable isosceles triangles; we will get two nonagons,whichwill not be congruent to each other. (In Fig. 9we can see the two nonagons P andQ, which are non-congruent parts of a regular dodecagon.) We might observe that thesupport functions hP and hQ of P and Q have the property that {hP(u), hP (−u)} ={hQ(u), hQ(−u)}. Hence wP (u) = wQ(u) for all u; furthermore, the width functionis a rotated copy of the brightness function; thus this example has the two requiredproperties.

Using these nonagons P and Q, we can construct a 3-dimensional pair of polyhedrawith the same brightness function and volumes. Consider the prisms HP := P × Iand HQ := Q × I , where I is a segment orthogonal to the common plane H of P andQ. By the following theorem, these prisms have the required properties.

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Theorem 6 Let P and Q be convex bodies of dimension n with the same brightnessfunction and volume. Let I be a segment orthogonal to that subspace H of the (n+1)-dimensional Euclidean space, which contains P and Q. Then the prisms HP and HQ

also have the same brightness function and (n + 1)-dimensional volume.

Proof The equality voln+1(HP ) = voln+1(HQ) is obvious.First we prove that the orthogonal projections P ′ and Q′ of P and Q to a subspace

� with the unit normal vector u have the same Hausdorff measure. (In the case when� is not orthogonal to H , the projection has a non-zero n-dimensional volume, whichagrees with the measure above, and in the orthogonal case this measure is equal tothe (n − 1)-dimensional volume of the projection.) Let the unit normal vector ofH be h and I = αh. Since P and Q have the same n-dimensional volume, we havevoln(P ′) = |〈u, h〉| voln(P) = |〈u, h〉| voln(Q) = voln(Q′). If it is non-zero, then theprojections have the same n-dimensional volume. If it is zero, H and� are orthogonalto each other, and the projections of P ′ and Q′ are the respective shadows of P andQ on an (n − 1)-dimensional subspace of H . Since their brightness functions agree,the (n − 1)-dimensional volumes of P ′ and Q′ are equal to each other, as we stated.

Assume that H and � are not orthogonal. Then the projection (HP |u⊥) is theconvex hull of the projections P ′ and P ′ + (αh − 〈u, αh〉u) of the polytopes P andP + αh of dimension n, where + denotes the vector sum. Hence by Eq. (12)

voln(HP |u⊥) = voln(P′) + |α||(h − 〈u, h〉u)| voln−1(P

′|(h − 〈u, h〉u)⊥).

On the other hand, we have

(P ′|(h − 〈u, h〉u)⊥) = ((P|�)|(h − 〈u, h〉u)⊥) = (P|H ∩ �),

since 〈v, u〉 = 0 and 〈v, h − 〈u, h〉u〉 = 0 ensures 〈v, h〉 = 0. Thus

voln(HP |u⊥) = voln(P′) + |α||(h − 〈u, h〉u)| voln−1(P|H ∩ �)

= voln(Q′) + |α||(h − 〈u, h〉u)| voln−1(Q|H ∩ �) = voln(HQ |u⊥),

as we stated.If H and � are orthogonal, then

voln(HP |u⊥) = voln−1(P|H ∩ �) = voln−1(Q|H ∩ �) = voln(HQ |u⊥)

immediately gives the required result. ��Using Theorem 6 we get an example in arbitrary dimension for non-congruent

convex bodies with the same brightness function and volume.The following problem leads to some interesting relations between volume func-

tions.

Definition 2 (Horváth and Lángi 2014) If for a convex body K ⊂ Rn , voln(conv(K ∪

(v+K ))) has the same value for any non-overlapping translate v+K of K , which hasa common point with K , then we say that K satisfies the translative constant volumeproperty.

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On Convex Bodies that are Characterizable. . . 17

If K is centered and satisfies the translative constant volume property, then GK (x)depends only on the norm ‖x‖K of x . In fact, if K is centered then it defines a norm‖·‖K , andv+K touches K if and only if‖v‖K = 2.Let v = αu; then‖v‖K = |α|‖u‖Kwhere u is a Euclidean unit vector. From the translative constant volume property, itfollows that

const. = GK (αu) = voln(K ) + |α| voln−1(K |u⊥)

= voln(K ) + ‖v‖K‖u‖K voln−1(K |u⊥) = voln(K ) + 2

voln−1(K |u⊥)

‖u‖K .

This implies that voln−1(K |u⊥)‖u‖K is constant for Euclidean unit vectors, and hence the

value of GK (αu) depends only on the value of ‖αu‖K , as we have claimed. Hence thequestion which bodies satisfy the translative constant volume property is analogous tothe question of Meyer et al. (1993) on covariogram stated in 1993. They proved thatif K is a centred body such that gK (x) depends only on the norm ‖x‖K of x , then Kis an ellipsoid.

We recall that a 2-dimensional o-symmetric convex curve is a Radon curve if, forthe convex hull K of a suitable affine image of the curve, it is true that K ◦ is a rotatedcopy of K by

π

2(cf. Martini and Swanepoel 2006). The concept of a Radon curve was

also raised in connection with the concept of Birkhoff orthogonality. Radon gave thefirst example in a Minkowski plane, in which the Birkhoff orthogonality relation is asymmetric one. On the other hand, it can be proved that, in a higher dimension, theBirkhoff orthogonality is a symmetric relation if and only if the norm is Euclidean,e.g. the unit ball of the space is an ellipsoid. The long history of this nice conceptcan be found in Sect. 4.7 of the book Thompson (1996). We also suggest to read theoriginal work of Radon (1916). In the paper Horváth and Lángi (2014), we can findthe following theorem:

Theorem 7 (Horváth and Lángi 2014) For any disk (planar convex body) K , the fol-lowing properties are equivalent.

(1) K satisfies the translative constant volume property.(2) The boundary of the central symmetral of K is a Radon curve.(3) K is a body of constant width in the Radon norm.

This statement motivates a new conjecture, since it is known [cf. Alonso and Benítez(1989) or Martini and Swanepoel (2006)] that for d ≥ 3, if every planar section of anormed space is Radon, then the space is Euclidean; that is, its unit ball is an ellipsoid.

Conjecture 5 (Horváth andLángi 2014)Let d ≥ 3. If some centrally symmetric convexbody K ⊂ R

d satisfies the translative constant volume property, then K is an ellipsoid.

If K and K + tp are touching at the point p ∈ K , then p − tp ∈ K and from Eq.(12) we get that GK (tp) = vol(K ) + |tp| voln−1(K |t⊥p ). The chord [p, p − tp] ofK is a so-called affine diameter of K . The general properties of affine diameters aredescribed in the survey of Soltan (2005). In the centrally symmetric case, we know thata chord containing the center is an affine diameter. These affine diameters determine

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18 A. G. Horváth

the radial function of the body K . If the origin x is the center of K , then we definethe radial function by the equality:

ρK (u) := sup{t ∈ R : tu ∈ K }. (13)

The radial function can be considered for all convex bodies such that the origin is aninterior point of K . There is the following relation between the support function andthe radial function:

ρK (u) = 1

hK ◦(u),

where K ◦ is the polar body of K . The polar projection body of K is the body�◦K :=�K ◦. An old problem is the so-called polar projection problem; it was raised by Petty(1971) and is mentioned by Gruber (1987) and Lutwak in Lutwak (1993).

Conjecture 6 (Petty 1971) If some centrally symmetric convex body K ∈ Rd satisfies

�◦K = λK, then K is an ellipsoid.

This problem has a form also in the non-symmetric case, asking which bodies havethe property that their projection bodies and difference bodies are polars of each other.Martini (1991) proved that the only polytope with this property is the simplex. Thereasonwhywementioned here the polar projection problem is that it is equivalent to theconjecture on translative constant volume property. (We will prove it in a forthcomingpaper.)

4 Remarks on the Homothetic Versions of the Above Problems

First of all, I would like to mention the paper of Meyer et al. (1993) in which we canfind the following theorem:

Theorem 8 (Martini 1991) Let K ⊂ Rn be a centrally symmetric convex body. If for

some τ the volume voln(K ∩ {τK + x}) depends only on the Minkowski norm ‖x‖K ,then K is an ellipsoid.

For the convex-hull function, an analogous result was proven by Castro (2015). Heproved the following two statements:

Theorem 9 (Castro 2015) Let K ⊂ Rn be a convex body with the origin o in its

interior. If there is a number λ ∈ (0, 1) such that voln conv{K ∪ (λK + x)} dependsonly on the Euclidean norm ‖x‖, then K is a Euclidean ball.

Theorem 10 (Castro 2015) Let K ⊂ Rn be a convex body with the origin o in its

interior, and let L ⊂ Rn be a centrally symmetric convex body centered at the origin.

If there is a number λ ∈ (0, 1) such that voln conv{K ∪ (λK + x)} depends only onthe Minkowski norm ‖x‖L , then L is homothetic to K .

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On Convex Bodies that are Characterizable. . . 19

Unfortunately, the witty proofs of these statements cannot be applied to the case ofConjecture 5. However, these results suggest the following problem.

Definition 3 Denote by

GK ,λ(t) := voln conv {K ∪ (λK + t)}

the λ-homothetic convex-hull function of K .

Problem 4 Does the λ-homothetic convex-hull function GK ,λ(t) determine the bodyK? What can we say about K if we know more λ-homothetic convex-hull functions ofit?

Acknowledgements Open access funding provided by Budapest University of Technology and Economics(BME).

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